Approximation problems for dynamic equations on time scales , bài toán xấp xỉ cho phương trình động lực trên thang thời gian 624601

VIETNAM NATIONAL UNIVERSITY, HANOIHANOI UNIVERSITY OF SCIENCE Nguyen Thu Ha APPROXIMATION PROBLEMS FOR DYNAMIC EQUATIONS ON TIME SCALES Speciality: Differential and Integral Equations Sp

VIETNAM NATIONAL UNIVERSITY, HANOI

HANOI UNIVERSITY OF SCIENCE

Nguyen Thu Ha

APPROXIMATION PROBLEMS FOR DYNAMIC

EQUATIONS ON TIME SCALES

THESIS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

HANOI 2017

VIETNAM NATIONAL UNIVERSITY, HANOI

HANOI UNIVERSITY OF SCIENCE

Nguyen Thu Ha

APPROXIMATION PROBLEMS FOR DYNAMIC

EQUATIONS ON TIME SCALES

Speciality: Differential and Integral Equations Speciality Code: 62 46 01 03

THESIS FOR THE DEGREE OFDOCTOR OF PHYLOSOPHY IN MATHEMATICS

Supervisor: PROF DR NGUYEN HUU DU

HANOI 2017

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Nguy„n Thu H

LÜC TR N THANG TH˝I GIAN

Chuy¶n ng nh: Ph÷ìng tr…nh Vi ph¥n v T‰ch ph¥n M¢ sŁ: 62 46 01 03

Ng÷íi h÷îng d¤n khoa håc:

GS TS NGUY N HÚU D×

H N¸I 2017

1.5.2 Exponential stability of linear dynamic equations with

con-1.6 Hausdor distance ffChapter 2 On the convergence of solutions for dynamic equations

on time scales

2.1 Time scale theory in view of approximative problems 2.2 Convergence of solutions for -dynamic equations on time scales 362.2.1

3.2.2 Solution of linear implicit dynamic equations with constant

3.2.3 Spectrum of linear implicit dynamic equations with constant

ii

3.3 Data-dependence of stability radii

3.3.1 Stability radius of linear implicit dynamic equations 3.3.2 Data-dependence of stability radii Conclusion

The author’s publications related to the thesis

Bibliography

First and foremost, I want to express my deep gratitude to Prof Dr NguyenHuu Du for accepting me as a PhD student and for his help and advice while Iwas working on this thesis He has always encouraged me in my work andprovided me with the freedom to elaborate my own ideas

I also want to thank Dr Do Duc Thuan and Dr Le Cong Loi for all the help theyhave given to me during my graduate study I am so lucky to get their support

I wish to thank the other professors and lecturers at Faculty of Mathematics, chanics and Informatics, Hanoi University of Science for their teaching, continuoussupport, tremendous research and study environment they have created I alsothank to my classmates for their friendship and suggestion I will never forget theircare and kindness Thank you for all the help and making the class like a family

Me-Last, but not least, I would like to express my deepest gratitude to my family.Without their unconditional love and support, I would not be able to do what Ihave accomplished

Hanoi, December 27, 2017

PhD student

Nguyen Thu Ha

iv

The characterization of analysis on time scales is the unification and expansion ofresults obtained on the discrete and continuous time analysis In some lastdecades, the study of analysis theory on time scales leads to much more generalresults and has many applications in diverse fields One of the most importantproblems in analysis on time scales is to study the quality and quantity of dynamicequations such as long term behaviour of solutions; controllability; methods forsolving numer-ical solutions In this thesis we want to study the analysis theory ontime scales under a new approach That is, the analysis on time scales is also anapproxima-tion problem Precisely, we consider the distance between the solutions

of different dynamical systems or study the continuous data-dependence of somecharacters of dynamic equations

The thesis is divided into two parts Firstly, we consider the approximation problem

to solutions of a dynamic equation on time scales We prove that the sequence ofsolutions xn(t) of dynamic equation x = f(t; x) on time scales fTng1n=1 converges tothe solution x(t) of this dynamic equation on the time scale T if the sequence ofthese time scales tends to the time scale T in Hausdorff topology Moreover, wecan compare the convergent rate of solutions with the Hausdorff distance between

Tn and T when the function f satisfies the Lipschitz condition in both variables.Next, we study the continuous dependence of some characters for the linearimplicit dynamic equation on the coefficients as well on the variation of time scales.For the first step, we establish relations between the stability regionscorresponding a sequence of time scales when this sequence of time scalesconverges in Hausdorff topology; after, we give some conditions ensuring thecontinuity of the spectrum of matrix pairs; finally, we study the convergence of thestability radii for implicit dynamic equations with general structured perturbations

on the both sides under the variation of the coefficients and time scales

Tâm t›t

°c tr÷ng cıa gi£i t‰ch tr¶n thang thíi gian l thŁng nh§t v mð rºng c¡c nghi¶ncøu ¢ ⁄t ÷æc Łi vîi gi£i t‰ch tr¶n thíi gian li¶n töc ho°c thíi gian ríi r⁄c Trong c¡cth“p ni¶n vła qua, vi»c nghi¶n cøu lþ thuy‚t gi£i t‰ch tr¶n thang thíi gian cho tanhi•u k‚t qu£ tŒng qu¡t v câ nhi•u øng döng v o c¡c l¾nh vüc kh¡c nhau Mºttrong nhœng b i to¡n quan trång cıa gi£i t‰ch tr¶n thang thíi gian l nghi¶n cøu t

‰nh ch§t ành t‰nh v ành l÷æng cıa ph÷ìng tr…nh ºng lüc Trong lu“n ¡n n y,chóng tæi muŁn nghi¶n cøu lþ thuy‚t gi£i t‰ch tr¶n thang thíi gian theo c¡ch ti‚pc“n mîi â l gi£i t‰ch tr¶n thang thíi gian cÆn l b i to¡n x§p x¿ Cö th” hìn, chóngtæi x†t kho£ng c¡ch giœa c¡c nghi»m cıa c¡c h» ºng lüc kh¡c nhau v s‡ nghi¶ncøu sü phö thuºc li¶n töc cıa mºt sŁ °c tr÷ng cıa ph÷ìng tr…nh ºng lüc theo dœli»u cıa ph÷ìng tr…nh

Lu“n ¡n bao gçm hai phƒn ch‰nh Tr÷îc h‚t, chóng tæi x†t b i to¡n x§p x¿ nghi»mcıa ph÷ìng tr…nh ºng lüc tr¶n thang thíi gian v chøng minh ÷æc d¢y c¡c nghi»m

xn(t) cıa ph÷ìng tr…nh x = f(t; x) tr¶n d¢y thang thíi gian t÷ìng øng fTng1n=1 s‡ hºi

tö ‚n nghi»m x(t) cıa ph÷ìng tr…nh n y tr¶n thang thíi gian T n‚u nh÷ d¢y thang thíigian n y hºi tö v• thang thíi gian T theo kho£ng c¡ch Hausdorff Hìn nœa, chóng tæicông ¡nh gi¡ ÷æc tŁc º hºi tö cıa c¡c nghi»m theo tŁc º hºi tö cıa d¢y thang thíi giankhi h m f thäa m¢n i•u ki»n Lipschitz theo c£ hai bi‚n

Ti‚p theo, ta nghi¶n cøu sü phö thuºc theo tham sŁ v theo sü bi‚n thi¶n cıa thangthíi gian cıa mºt sŁ °c tr÷ng cıa ph÷ìng tr…nh ºng lüc 'n tuy‚n t‰nh B÷îc ƒuti¶n, ta thi‚t l“p mŁi li¶n h» giœa c¡c mi•n Œn ành t÷ìng øng cıa d¢y c¡c thangthíi gian khi d¢y thang thíi gian n y hºi tö theo tæ pæ Hausdorff CuŁi còng,chóng ta nghi¶n cøu sü hºi tö cıa b¡n k‰nh Œn ành cıa ph÷ìng tr…nh ºng lüc 'ntuy‚n t‰nh chàu nhi„u c§u tróc ð c£ hai v‚ cıa ph÷ìng tr…nh khi c£ h» sŁ vthang thíi gian •u bi‚n thi¶n

vi

This work has been completed at Hanoi University of Science, Vietnam NationalUniversity under the supervision of Prof Dr Nguyen Huu Du I declare hereby thatthe results presented in it are new and have never been used in any other thesis

Author:

Nguyen Thu Ha

List of Figures

1.1 Points of the time scale T 12

2.1 The graph of the solution xn(t) on the time scale Tn 49

2.2 The graph of the solution x4(t) on the time scale T4 50

2.3 The graph of the solution xn(t) on the time scale Tn 55

viii

N; Q; R; C The set of natural, rational, real, complex numbers

R(Tk; X) The set of regressive functions, defined on T

and take the value on X

R+(Tk; R) The set of positive regressive function defined on T

and taking values in R

UT = UT(T) The uniformly exponential stability domain of time

scale T

(A) The set of the eigenvalues of the matrix A

(A; B) The set of solutions of det( A B) = 0

S(T) Exponential stability domain of time scale T

sup; inf supremum, infimum

x

The theory of ordinary differential equations (ODEs for short) is a theoretical sys-tem,rather academic but going into the practical issues In practical problems, ordinarydifferential equations occur in many scientific disciplines, for instance in physics,chemistry, biology, and economy Under the name "modeling", the ordinary differentialequation is a useful tool to describe the evolution in times of population

Therefore, studying the qualitative and quantitative properties of differential equa-tions

is important both in theory and practice For the qualitative properties, the long termbehavior of the solutions has got many interests The main tools in study-ing thestability are Lyapunov functions, Lyapunov exponents or spectral analysis of matrices.For the quantitative analysis, we have to find numerical approximations to thesolutions since almost ODEs can not be solved analytically The Euler methods arewell-known because it is simple and useful, see [9, 19, 22, 63]

Besides, the theory of difference equations has a long term of history Right from thedawn of mathematics, it was used to describe the evolution of a lapin population withthe name "Fibonacci sequence" Difference equations might define the simplestdynamical systems, but nevertheless, they play an important role in the investigation

of dynamical systems The difference equations arise naturally when we want to studymathematical models describing real life situations such as queuing problems,stochastic time series, electrical networks, quanta in radiation, genetics in biology,economy, psychology, sociology, etc., on a fixed period of time They can also beillustrated as discretization of continuous time systems in computing process

On the other hand, in some last decades, the theory of time scales, under thename Analysis on time scales, was introduced by Stefan Hilger in his PhD thesis(supervised by Bernd Aulbach) in order to unify continuous and discrete analy-ses

As soon as this theory was born, it has been received a lot of attention, see [6, 7,

13, 39, 41, 43] Until now, there are thousands of books and articles dealt with thetheory of analysis on the time scale Many familiar results concerning to

qualitative theory such as stability theory, oscillation, boundary value problem incontinuous and discrete times were "shifted" and "generalized" to the time scale.

Studying the theory of time scales leads also to some important applicationssuch as in the study of insect density model, nervous system, thermalconversion process, the quantum mechanics and disease model

One of the most important problems in the analysis on time scales is to investigatedynamic equations Many results concerning differential equations carry over quiteeasily to corresponding results for the difference equations, while other resultsseem to be completely different in nature from their continuous counterparts Thestudy of dynamic equations on time scales gives a better perspective and revealssuch dis-crepancies between the differential equations and difference equations.Moreover, it helps us avoid proving results twice, one for differential equations andone for dif-ference equations The general idea is to prove results for dynamicequations where the domain definition of the unknown function is a so-called timescale T, which is an arbitrary nonempty closed subset of real numbers R Bychoosing the time scale to be the set of real numbers R, the general result yields aresult concerning an ordinary differential equation as studies in a first course indifferential equations On the other hand, if the time scale is the set of integers Z,the same general result yields a result for difference equations

However, studying the theory of dynamic equations on time scales leads tomuch more general results since there are many more other complex timescales than the above two sets That is why so far the analysis on time scales isstill an attracted topic in mathematical analysis Especially, there are still manyopen problems in the studying dynamic equations on time scales

The aim of this thesis is to consider the analysis on time scales under a new point ofview That is not only a unification, but also in the view of approximation Pre-cisely,

we want to consider the distance between the solutions of different dynamical systems

or to study the continuous dependence of some characters of dynamics equa-tions ondata such as spectrum, stability radii when both the coefficients and time scales vary.The two following topics will be dealt with in this thesis:

1. Approximation of solutions

We begin firstly by analyzing the Euler method for solving the stiff initial valueproblem It is known that there are not many classes of ordinary differential equa-tions for which we can represent explicitly their solutions via analysis formulas,especially for nonlinear differential equations Therefore, finding numerical solu-

2

tions of differential equations plays an important role in both theory and practice.

So far one proposes many algorithms to solve numerically solutions of adifferential equation Among these algorithms, we have to take into account theEuler explicit and implicit methods Let us consider the initial value problem (IVP)

8

<x(t) = f(t; x(t));

(0.1):x(t0) = x0:

The approximation of the solution x(t) of (0.1) will carry out at some differentvalues of times, say mesh points, on the interval [t0; T ] To do that, for every n 2

N, we consider a partition of [t0; T ] consisting of points

Our problem is to give conditions of the function f and the partition (0.2)ensuring the convergence of explicit Euler method:

sup x(n)

k j

k

These conditions can be found in [19, 22, 44]

Although the explicit Euler method is quite simple and easy to implement, even wecan carry out by portable calculators, and we can show the convergence of thismethod However, it has accumulation error in the processes of calculation andEuler scheme can also be numerically unstable, especially for stiff equations sinceexplicit method requires impractically small time steps to keep the error in theresult bounded and the convergent rate is not good Therefore, one deals with thesecond Euler method, called Euler implicit method or Euler backward method Inthis method, instead of the equation (0.3), we consider the difference equation

x(0n) = x0; x(i+1n) = x(in) + (t(i+1n) t(in))f(t(i+1n); x(i+1n)); i = 0; 1; : : : ; kn 1: (0.5)

This differs from the Euler explicit method in that the latter uses f(t(kn); x(kn)) in place off(t(i+1n); x(i+1n)) In Euler implicit method, the new approximation x(i+1n) appears onboth sides of the equation (0.5) and thus the method needs to solve an algebraic

to use the Euler implicit method because many problems arising in practice arestiff and it can achieve a higher convergent rate The error at a specific time t isO(h) (using the big O notation) and the Euler implicit method is in generalunconditionally stable For a stiff problem, explicit methods need to take verysmall time steps Therefore, meanwhile the calculation is complicated, the Eulerimplicit method could be preferred in this case (see [19, 22, 44]).

We now consider these two methods in different way That is, when onediscretises the dynamic equation (0.1) on the real line [0; T ] by Euler explicitmethod, we obtain the difference equation (0.3) On the time scale languages,this equation is in fact the dynamic equation x (t) = f(t; x(t)) on time scale Tn

described by (0.2) Similarly, the equation (0.5) is the dynamic equation xr(t) = f(t;x(t)) on Tn When the mesh steps of Euler methods tend to 0, the sequence oftime scales Tn converges to T in some senses and the convergence of Eulermethods means the convergence of the sequence of solutions x((n)), the solution

of above equations on the time scales Tn

Therefore, for the first part of this thesis, we follow this idea to set up an imation problem in a more general context: Let T be a time scale and let fTng1n=1

approx-be a sequence of time scales, which converges to T in some senses Considerthe dynamic equation

8x (t)

< x(t 0 )

:where either t 2 T or t 2 Tn Assume that on every time scale Tn (resp T), theCauchy problem of the equation (0.7) has a unique solution xn(t) (resp x(t)).Then, the question here is whether we can specify conditions to have

xn( ) ! x( ) as n ! 1:

Further, how can we estimate the rate of this convergence?

4

We can also deal with a similar problem for the nabla dynamic equations on time scales That is instead of considering the equation (0.7), we study the equation

and put similar questions on

2. Continuous dependence of the spectrum and stability radius on data

The second topic dealt with in this thesis is to consider the data dependence of thespectrum and stability radius for linear implicit dynamic equations on time scales

In the recent years, several technical problems in electronic circuit theory androbotic designs, chemical engineering, etc., see [15, 16, 28, 55] lead to theproblem of inves-tigating the dynamic implicit equation (IDEs for short)

f(X (t); X(t)) = 0;

where the leading term X can not be explicitly solved from X(t) The linear form

of this equation (LIDEs for short) is

AX (t) BX(t) = 0;

where A and B are two constant matrices (see [24, 46]) According to [24] and[58], the investigation of the so-called index of the pencil of matrices fA; Bg isnecessary but the situation becomes more complicated Note that if A is anonsingular matrix, then equation (0.11) can easily be reduced to an ordinarydifferential equation by multiplying both sides of (0.11) by A 1 In this case, it isknown that if the original equation (0.11) is exponentially stable then it is stillstable under sufficiently small perturbations In general case where A may bedegenerate, this property is no longer valid for the equation (0.11) since thestructure of the solutions of a LIDEs depends strongly on the index of fA; Bg(see [31, 45, 46, 58]) and the solutions of (0.11) contain several components,which are related by algebraic relations Under perturbations, the index of thesystem might be changed, which leads to the change of the algebraic relations

In view of spectral theory, it is known that the uniformly exponential stability hasclose relations with the spectrum (A; B) of the matrices pencil fA; Bg, i.e., theset of roots of the equation det( A B) = 0: The changing in parameters of indexcauses, without additional assumptions, the sharp change of the spectrum (A;B) and the continuity of spectrum on the data is no longer valid

Then, the question whenever the spectrum (A; B) depends continuously in fA;

Bg is important in both theory and practice Thus, we come to the following

problem on time scales

Problem: consider a family of linear implicit dynamic equations on the time

scales T

n

Anx n (t) = Bnx(t); n 2 N;

with An; Bn 2 Cm m Which conditions ensure that the system Ax (t) = Bx(t) on the

time scale T is exponentially stable if the system (0.12) is exponentially stable

for every n and lim (An; Bn; Tn) = (A; B; T)?

n!1

In parallel, we have a similar problem for stability radii of implicit dynamic

equa-tions It is well-known that if the trivial solution x 0 of the linear differential

system x0 = Bx (resp difference system xn+1 = Bxn) is exponentially stable,

then for a small perturbation , the system

x0 = (B + D E)xand respectively,

xn+1 = (B + D E)xn;

is still exponentially stable Where is an unknown disturbance matrix and D; E are

known scaling matrices defining the structure" of the perturbation The question rises

here: how large is possible in order to keep the stability of the system (0.13) (resp

(0.14)) The threshold between the stability and instability, which measures the stability

robustness of system to such perturbation, is called the stability radius It is defined as

the smallest (in norm) complex or real perturbations destabilizing the equation If

complex perturbations are allowed, this measure is called the complex stability radius,

if only real perturbation are considered the real radius is obtained The concept of

stability radii was introduced by Hinrichsen and Pritchard [48] in 1986 for linear

time-invariant systems of ordinary differential equations with respect to time-time-invariant input,

i.e., static perturbations In this work, authors have shown that the complex stability

radius of linear differential equation (0.13) is given by

max

Since then, this problems have been getting a lot of attentions from many research

groups of mathematics in the world D Hinrichsen and N.K Son [52] (1989) have

investigated the difference equation with the perturbation of the form xn+1 = (B + D

E)xn and have shown that the complex stability radius is computed by the

6

Ax (t) = Bx(t);

subject to the perturbations

~ ~[A;B]=[A;B]+D E =[A+D E1

where A; B 2 Cm m, D 2 Cm l; E1; E2 2 Kq m; E = [E1; E2], the perturbation

2 Cl q With these perturbations, the system (0.19) leads to

We emphasise that the form (0.20) says that both sides of (0.19) are perturbed by

As far as we know, the perturbation on the left side of (0.19) is very sensitive since

it can make its index change roughly Following the analysis in [68], the stability

7

Du-Lien-Linh for the first time in [36]; Du-Linh [32] and Du-Linh [34] considerthis continuous dependence under the statement of small parameters Theyclaim that if r(E + "F; A; B; C) is the stability radius of

(E + "F )x0 = (A + B C)x;

Then, under some heavy assumptions, it is shown that

lim r(E + "F; A; B; C) = minfr(E; A; B; C); r(F22; A22; B2; C2)g;

on the data of the stability radii for implicit dynamic equations (0.23) when (An;

Bn; Tn) converges

The content of this thesis are as follows Chapter 1 presents some basic knowledgeabout time scales such as the definition of derivative, integration on time scales.Besides, we give concepts of the exponential function, exponential stability region aswell as some results of the stability for the dynamic equations on time scales

The second chapter is devoted to the study of the convergence of solutions to namic equations We endow the set of time scales with the Hausdorff distance Let

dy-xn(t) (resp x(t)) be the solution of the equation x = f(t; x) (or the nabla dynamicequation xr = f(t; x)) on time scale Tn (resp on T) Under the assumption that f(t; x)satisfies the Lipschitz condition in the variable x and the sequence of time scales

fTng1n=1 converges to the time scale T in the Hausdorff topology, Theorem 2.2.7shows that lim xn(t) = x(t) Moreover, in case f satisfies the Lipschitz con-

considered as a novel approach to the convergence problems of the

approximative solutions

In the last chapter, we study some problems concerning the stability regions,

the spectrum of matrix pairs, the exponential stability and their robustness

measure for linear implicit dynamic equations of arbitrary index (0.19) subjected

to general structured perturbations of the form

[Aen; Ben] [An; Bn] + Dn nEn;

where n; n 2 N; are unknown disturbance matrices; Dn; En are known scaling matrices

defining the structure" of the perturbations On each time scale Tn, the stable radius of

this system were defined by the formula (0.22) Some characteristics of the stability

regions corresponding to a convergent sequence of time scales are derived In more

details, Theorem 3.1.7 tells us that the stability regions depend continuously on the

time scales Concretely, if Tn tends to T in Hausdorff topology

1

3.2.11 showsSthat T

all n 2 N, then we have n lim (A n ; B n ) = (A; B) in the Hausdorff distance Further,

Proposition 3.2.13 tells us that in case Ind(A; B) > 1 and

the se propositions we ca n claim tha t the sta bility radii r(A

En; Tn) are lower semi continuous in An; Bn; Dn; En; Tn By Theorems 3.3.6 and

3.3.8 we see that with some further conditions, if lim (An; Bn; Dn; En; Tn) = (A; B; D;

n!1

E; T) then

r(A; B; D; E; T) = lim inf r(An; Bn; Dn; En; Tn):

n!1

In conclusion, we think that the theory of dynamic systems on an arbitrary time

scale was found promising because it demonstrates the interplay between the

theories of continuous-time and discrete-time systems, see [7, 27, 43, 47, 65] It

enables to analyze the stability of dynamical systems on non-uniform time

domains which are subsets of real numbers [66] Based on this theory, stability

analysis on time scales has been studied for linear time-invariant systems [61],

linear time-varying dynamic equations [26], implicit dynamic equations [39, 68],

switched systems [66, 67] and finite-dimensional control systems [10, 29]

Therefore, it is meaningful to investigate the dependence of stability

characteristics of these systems on time scales and coefficients as well

Parts of the thesis have been published in

9

1. Nguyen Thu Ha, Nguyen Huu Du, Le Cong Loi and Do Duc Thuan (2016),

"On the convergence of solutions to dynamic equations on time scales",Qual Theory Dyn Syst., 15(2), pp 453 469

2. Nguyen Thu Ha, Nguyen Huu Du, Le Cong Loi and Do Duc Thuan (2015),

"On the convergence of solutions to nabla dynamic equations on timescales", Dyn Syst Appl., 24(4), pp 451-465

3. Nguyen Thu Ha, Nguyen Huu Du and Do Duc Thuan (2016), "On dependence of stability regions, exponential stability and stability radii forim-plicit linear dynamic equations", Math Control Signals Systems, 28(2),

data-pp 13-28

10

Chapter 1

Preliminary

In this chapter we survey some basic notions related to the analysis on timescales We also introduce the so-called cylinder transformation order to define theuniform stability region of a time scale and the uniformly exponential stability forthe dy-namic equations on time scales The details for these concepts anddefinitions can be referred to the famous papers of S Hilger [43], C Potzsche, S.Siegmund and F Wirth [61] and the book of M Bohner and A Peterson [12]

1.1 Definition and example

A time scale T is an arbitrary nonempty closed subset of the real numbers R

We assume throughout of this thesis that the time scale T has the topologyinherited from the standard topology of real numbers R

On the time scale T, we define the forward jump operator and backward jumpoper-ator as follow

1. Forward jump operator: given by (t) := inffs 2 T : s > tg, t 2 T

2. Backward jump operator: defined by (t) := supfs 2 T : s < tg, t 2 T

In this definition we put inf ; = sup T; sup ; = inf T (i.e., (t) = t if t = max T; and (t)

Definition 1.1.1 On time scale T the forward graininess function : T ! R+ isdefined by (t) := (t) t; t 2 T and the backward graininess (t) := t (t); t 2 T

The left dense or right dense is illustrated by the Figure 3.1.1

Figure 1.1: Points of the time scale T.

Denote (a; b)T = ft 2 T : a < t < bg: Similarly, we can denote (a; b]; [a; b) or [a;b] To simplify notations, from now on we write (a; b); (a; b]; [a; b); [a; b] instead

of (a; b)T; (a; b]T; [a; b)T; [a; b]T if there is no confusion

If T has a left-scattered maximum Tmax, then define T = T n fTmaxg, otherwise T

= T Similarly, If T has a right-scattered minimum Tmin, then T = T n fTming,otherwise T = T

For any t 2 T we write f (t) for f( (t))

We give some examples of typical time scales

Let a; b > 0 be fixed real numbers We define the time scale Pa;b

by Pa;b = [1k=0[k(a + b); k(a + b) + a]:

12

In this case,

8

<t(t) =

if t 2 [1k=0[k(a +b); k(a + b) + a); if t

2 [1k=0fk(a+ b) + ag:

Let q > 1, we define the time scale qZ by

1

H 0 = 0; H n =

by natural way However, thereare some kinds of points in atime scale, we introduce somefurther notions of the continuity.

Definition 1.2.1

1. A function f : T! R is calledregulated, provided itsright-sided limits exist(finite) at all right-densepoints in T and its left-sided limits exist (finite) atall left-dense points in T

13

2. A function f : T ! R is said to be rd-continuous (right-dense continuous) if it

is continuous at right-dense points in T and its left-sided limits exist (finite)

at all left-dense points in T

3. A function f : T ! R is ld-continuous (left-dense continuous) if it is tinuous at left-dense points in T and its right-sided limits exist (finite) at allright-dense points in T

con-4. Let A be an m n matrix valued function on time scale T We say A( ) is continuous (resp ld-continuous) if each entry of A is rd-continuous (resp.ld-continuous) on T

rd-For two variables functions we have the following definition Let X be a Banach space

5 A mapping

(t; x) 7! f(t; x)

is said to be rd-continuous if it satisfies the following conditions

(a) f is continuous in (t; x) at each (t; x) where t is right-dense or t = max T;

Some results concerning rd-continuous and regulated functions are contained

in the following theorem

Theorem 1.2.2 Consider the function f : T! R, we have:

i) If f is continuous, then f is rd-continuous

ii) If f is rd-continuous, then f is regulated

iii) The jump operator is rd-continuous

iv) If f is regulated (rd-continuous), then so is f

v) Assume that f rd-continuous If g : T ! R is regulated (rd-continuous) then f

g has that property too

vi) f is a continuous function iff it is ld-continuous and rd-continuous at the

same time

1.2.2 Delta derivative

The derivative and difference operators are basic notions in analysis theory

There-fore, in order to unify them, we give a definition of the so-called delta (or

Hilger) derivative for functions defined on time scales

Definition 1.2.3 Assume f : T ! R is a function and let t 2 T Then we define the

-derivative of the function f at t to be a number f (t) (provided it exists) with the

property that given any " > 0, there is a neighbourhood U of t such that

The set of the -differentiable functions on T is denoted by

Crd1(T; Rd) = ff : T! Rd : f is differentiable and f is rd-continuousg:

Similar to the derivative concept of the real functions, we have the following

prop-erties

Theorem 1.2.4 Assume f: T ! R and let t 2 T Then we have the following

statements:

1 If f is differentiable at t then f is continuous at t

2 If f is continuous at t and t is right-scattered then fis differentiable at t with

exists as a finite number In this case

Next, we give the formulas of the derivative of sums, products, and quotients of

differentiable functions in the following theorem

Theorem 1.2.6 Assume f , g: T! R are differentiable at t 2 T Then:

1 The sum f + g: T! R is differentiable at t with

16

1.2.3 Nabla derivative

A twice born notion of delta derivative is nabla one It is generalized from thebackward difference equation xn xn 1 = f(n; xn)

A function f from T to R is called nabla regressive (respectively: positively

regres-sive) if 1 (t)f(t) =6 0 (respectively: 1 (t)f(t) > 0) for every t 2 T:

Definition 1.2.7 (Nabla derivative) A function f : T ! R is called nabla entiable at t if there exists a number fr(t) such that for all " > 0

differ-f( (t)) differ-f(s) fr(t)( (t) s)

jfor all s 2 U and for some neighbourhood U of t Then fr(t) is called the nabladerivative of f at t

If T = R then the nabla derivative is f0(t) from continuous calculus; if T = Z thenthe nabla derivative is the backward difference operator, rf(t) = f(t) f(t 1), fromdiscrete calculus In general, there is no relation between the delta and nabladerivatives

The nabla derivative has the similar properties as the delta one For the details,

we can refer to the book of M Bohner and A Peterson (2001) [12]

1.3 Delta and nabla integration

In this section we present some basic notions of the integral on time scales Thereare some ways to define the integral For example, we can define the Riemanndefinitive integral of a function by considering the partitions of the interval [t0; T ]into subintervals, the Darboux sums and then establishing the limit if it exists

In the following we introduce the basic definitions and properties about theLebesgue integral on time scales by Carath†odory extension For the details wecan refer to [42]

1.3.1 and r measures on time scales

Denote by A0 the family (collection) of all left closed and right open intervals of

T of the form [a; b) = ft 2 T : a 6 t < bg with a; b 2 T and a 6 b The interval [a; a)

is understood as the empty set A0 is a semi-ring of subsets of T

Let m1 : A0 ! [0; 1] be the set function on A0 (whose values belong to theextended real half-line [0; 1]) that assigns each interval [a; b) to its length b a.Then, m1 is accountably additive measure on A0 We introduce theCarath†odory extension m of m1 First, using the pair (A0; m1), we generate anouter measure m1 on the family of all subsets of T as follows Let E be anysubset of T If Vj 2 A0 (j = 1; 2; :::) is one finite or countable system of intervals in

T such that E [jVj, we say that Vj 2 A0 (j = 1; 2; :::) is a covering of E Insupposing that there is at least one covering of E, we put

m1(E) = inf m1(Vj) ;

j

where the infimum is taken over all coverings of E Second, we define the family A

= A(m1) of all m1-measurable subsets of T, consisting of all subsets A of T suchthat m1(E) = m1(E \ A) + m1(E \ Ac) holds for all E T, where Ac denotes thecomplement of A : Ac = T n A The family A(m1) of all m1 -measurable subsets of T

is a -algebra Third, we take the restriction of m1 to A(m1), which we denote by m This m (the Lebesgue -measure) is a countably additive measure on A(m1)

All intervals of family A0 including the empty set are -measurable By definition, it isobvious that the T is also -measurable Suppose that T has a finite maximum Tmax.Obviously, the set X = T n fTmaxg can be represented as a finite or countable union ofintervals of the family A0 and, therefore is -measurable Consequently, the single-pointset fTmaxg = T n X is -measurable as the difference of two mea-surable sets T and X.Evidently, the single-point set fTmaxg does not have a finite or countable covering byintervals of A0 Therefore, the single-point set fTmaxg and also any -measurable subset

of T containing Tmax have -measure infinity

Theorem 1.3.1 For each t0 2 TnfTmaxg, the single-point set ft0g is -measurableand its -measure is given by

m ft0g = (t0) t0Since each single-point subset of T is -measurable and since every kind of intervalcan be obtained from an interval of the form [a; b) by adding or subtracting the endpoints a and b, each interval of T is -measurable The following theorem givesformulas for evaluating the -measure of any interval of T

Theorem 1.3.2 If a; b 2 T and a 6 b, then

m [a; b) = b a; m (a; b) = b (a):

18

If a; b 2 T n fTmaxg and a 6 b, then

m (a; b] = (b) (a); m [a; b] = (b) a:

Now we define the concept of the Lebesgue r-measure on T Let B denote thefamily of all right closed and left open intervals of T of the form (a; b] = ft 2 T : a < t

6 bg with a; b 2 T and a 6 b The interval (a; a] is understood as the empty set B is

a semi-ring of subsets of T Further, let m2 : B ! [0; 1] be the set function on Bassigned to each interval (a; b] its length b a Then m2 is accountably additivemeasure on B We denote by mr the Carath†odory extension of the set function

m2 associated with the family B as above and call mr the Lebesgue r-measure

on T The following two theorems can be proved analogously to Theorems 1.3.1and 1.3.2

Theorem 1.3.3 For each t0

1.3.2 Integration

The Lebesgue integrals associated with the measures m and mr on T are calledLebesgue -integral and the Lebesgue r-integral on T, respectively For a measur-able set E T and a measurable function f : E ! R, we denote the integrals of

f over E corresponding to and r measures by

tively In case E = [a; b) we write

Ra

It is easy to see that any regulated function is integrable because it has at most

a countable number of discontinuous points In particular, all rd-continuous andld-continuous functions are both nabla and delta integrable

All theorems of the general Lebesgue integration theory, including theLebesgue dominated convergence theorem, will be held for the Lebesgue and rintegrals on T.

In general, there is no relation between the delta integral and nabla one.However, in case f is continuous we have

a b

Let T1 and T2 be two time scales Suppose that T1 T2 Let f be an integrable

function defined on the interval [a; b)T 1 Consider an extension fe of f to [a; b)T 2

20

as the following: for any t 2 [a; b]T 2 , put fe(t) = f(s) if t 2 [s; 1(s)], where 1 is

forward operator on T1 Then we have

Z

T

1.3.4 Polynomial on time scales

In this subsection, we start with the definition of the generalized polynomial,

which is given by

h0(t; ) = 1 and hk(t; ) = Zt

hk

for all t; 2 T and k = 1; 2; :::

The above definition obviously implies h1(t; ) = t

finding hk(t; ) is not easy in general But for a particular time scale it may be

easy to find these function For example, by induction, we will show that